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Properties

Label 2-1372-4.3-c0-0-4
Degree $2$
Conductor $1372$
Sign $0.781 + 0.623i$
Analytic cond. $0.684717$
Root an. cond. $0.827476$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)8-s + 9-s + 1.94i·11-s + (−0.222 − 0.974i)16-s + (0.900 − 0.433i)18-s + (0.846 + 1.75i)22-s − 1.56i·23-s − 25-s − 1.24·29-s + (−0.623 − 0.781i)32-s + (0.623 − 0.781i)36-s + 0.445·37-s − 0.867i·43-s + (1.52 + 1.21i)44-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)8-s + 9-s + 1.94i·11-s + (−0.222 − 0.974i)16-s + (0.900 − 0.433i)18-s + (0.846 + 1.75i)22-s − 1.56i·23-s − 25-s − 1.24·29-s + (−0.623 − 0.781i)32-s + (0.623 − 0.781i)36-s + 0.445·37-s − 0.867i·43-s + (1.52 + 1.21i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1372\)    =    \(2^{2} \cdot 7^{3}\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(0.684717\)
Root analytic conductor: \(0.827476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1372} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1372,\ (\ :0),\ 0.781 + 0.623i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.915100537\)
\(L(\frac12)\) \(\approx\) \(1.915100537\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 \)
good3 \( 1 - T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 - 1.94iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.56iT - T^{2} \)
29 \( 1 + 1.24T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.445T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 0.867iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.56iT - T^{2} \)
71 \( 1 - 0.867iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 0.867iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.937115686760187840894206042153, −9.248594847166064193556858655934, −7.77274878894390937841261791184, −7.10240648599961656956944271334, −6.42777720548225877206904851051, −5.27817135583631037551078744849, −4.43849193313729779426138301934, −3.93642090207885808573866183165, −2.43829543918284551206004420041, −1.65479257170231372030708350597, 1.70565747801894732789870220592, 3.22864169232827383148160280659, 3.77380907480696221800784992438, 4.87745400424388120671838243907, 5.83008141455236100331674650935, 6.31358204091571466139587615880, 7.52168740282223028389534751318, 7.894702867433736406253955734793, 8.977374239783769962513147399795, 9.812910958266163665458488483630

Graph of the $Z$-function along the critical line